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A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

%I A206289 #16 Nov 06 2014 08:04:09
%S A206289 1,1,2,4,10,25,73,214,679,2189,7331,24867,86269,302144,1072621,
%T A206289 3837768,13853674,50319789,183941789,675731105,2494370326,9244865453,
%U A206289 34394851701,128390336942,480749791772,1805161153783,6795744287172,25643914891284,96980809856731
%N A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).
%C A206289 Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).
%H A206289 Vaclav Kotesovec, <a href="/A206289/b206289.txt">Table of n, a(n) for n = 0..260</a>
%F A206289 G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
%F A206289 (1) G_n(x) = Series_Reversion( x*(1 - x^n) ),
%F A206289 (2) G_n(x) = x + x*G_n(x)^(n+1),
%F A206289 (3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).
%F A206289 a(n) ~ c * 4^n / n^(3/2), where c = 0.19197348199... . - _Vaclav Kotesovec_, Nov 06 2014
%e A206289 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
%e A206289 such that, by definition,
%e A206289 A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
%e A206289 where G_n( x*(1 - x^n) ) = x.
%e A206289 The first few expansions of G_n(x) begin:
%e A206289 G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
%e A206289 G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
%e A206289 G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
%e A206289 G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
%e A206289 G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
%e A206289 G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
%e A206289 Note that G_n(x) = x + x*G_n(x)^(n+1).
%o A206289 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
%o A206289 for(n=0,35,print1(a(n),", "))
%Y A206289 Cf. A206290, A194560.
%K A206289 nonn
%O A206289 0,3
%A A206289 _Paul D. Hanna_, Feb 05 2012